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May 26, 2005

Talking to myself

Posted by Robert H.

Since nobody answered my question about how to properly generalize the calibration condition for BPS-branes if the gauge field has curvature I have to do it myself.

Until this very minute, I have been preparing for today’s joint math/physics block seminar in Hamburg where we’re going to find out what Generalized Complex Geomtery is really about (Lubos has chatted about it in his reference frame). Not to arrive completely clueless I have been reading Gualtieri’s thesis which I can strongly recommend to everybody. It is an excelent read even for physicists! And there, in chapter 7 my question is answered:

You probably know that this generalized business works by considering the tangent and co-tangent bundles together. Then a generalized complex structure J is maps T⊕T* to itself and squares to −1 and fulfills some integrability condition. It’s easy to see that this condition contains complex, symplectic and Poisson geometry and interpolates between these. Furthermore it is co/invariant under transformations by closed 2-forms B and can be twisted by closed threeforms H, e.g. dB for not closed B.

Now consider a submanifold of this space on which there is a 2-form F with dF=H (0 without twist). The trick now is to look at the subbundle of T⊕T* on the submanifold such that the vector component X is tangent to the submanifold and the form component is given by iXF. The condition for this to be a generalized complex submanifold is now to require that this bundle is stable under J. And, as promised, this generalizes complex, Lagrangian and self-duality for F as BPS conditions. And there is also a spinorial description.

I must say, this story is one of those that is so beautiful that it can really foster your belief that there must be some truth to string theory!

May 18, 2005

String(n), Part II

Posted by Urs Schreiber

In the last entry I have listed some facts related to the group String(n). Here is the literature that my discussion was mainly based on as well as a review of what String(n) has to do with 2-groups and 2-bundles.

May 16, 2005

PSM and Algebroids, Part V

Posted by Urs Schreiber

Last time I discussed how the functorial definition of a p-bundle with p-connection can locally be differentiated to yield morphisms between p-algebroids. Now I think I have figured out the differential version of the transition law describing the transformation of these algebroid morphisms from one patch to the other. The result is a formalism that allows you to derive the infinitesimal cocylce relations of a nonabelian p-gerbe with curving and connection, etc. using just a couple of elementary steps.

May 13, 2005

PSM and Algebroids, Part IV

Posted by Urs Schreiber

Last time I discussed how Lie p-algebroids (and hence Lie p-algebras) and dg-algebras on graded vector spaces of maximal grade p are two aspects of the same thing. This goes a long way towards merging the study of p-bundles with p-connections with the study of algebroid morphisms as they arise in the Poisson σ-model, Dirac σ-models and other field theories.

May 4, 2005

PSM and Algebroids, Part III

Posted by Urs Schreiber

I have just returned from visiting Thomas Strobl at Jena University, where we talked about algebroids, gerbes, categorified gauge theory, and generalized geometry and how it all fits together. I have learned a lot in these discussions and have gotten a little closer to seeing the big picture, also thanks to the valuable pointers to the literature by Melchior Grützmann and Branislav Jurčo. Here I’ll list some useful and interesting facts – except for those that are top-secret…

(Please note that all my attributions in the following reflect only my level of awareness of the literature. I’d be grateful for corrections and further pointers to the literature.)

[Note: Users of non-Mac machines might have to download a new font in order to properly view all mathematical symbols in the following. More general information can be found here.]